On the regularity of the flow map for the gravity-capillary equations

Abstract

We prove via explicitly constructed initial data that solutions to the gravity-capillary wave system in ${\mathbb{R}}^3$ representing a 2d air-water interface immediately fails to be $C^3$ with respect to the initial data if the initial data $(h_0,{\psi }_0)\in H^{s+\frac{1}{2}}\otimes H^s$ for $s<3$. Similar results hold in ${\mathbb{R}}^2$ domains with a 1d interface. Furthermore, we discuss the illposedness threshold for the pure gravity water wave system.

31 pages, no figures; several errors in the Dirichlet-Neumann map expansion corrected thanks to a comment from Nicolas Burq, Thomas Alazard and Claude Zuilly

References in corpus (1)

• On the Water Waves Equations with Surface Tension

Cited by in corpus (1)

• On the well-posedness of a weakly dispersive Boussinesq system